Existence of positive solutions for a class of delay fractional differential equations with generalization to

$N$-term.

*(English)* Zbl 1217.34007
Summary: We established the existence of a positive solution of nonlinear fractional differential equations $\U0001d50f\left(D\right)[x\left(t\right)-x\left(0\right)]=f(t,{x}_{t}),t\in (0,b]$ with finite delay $x\left(t\right)=\omega \left(t\right),t\in [-\tau ,0]$, where $li{m}_{t\to 0}f(t,{x}_{t})=+\infty $, that is, $f$ is singular at $t=0$ and ${x}_{t}\in C([-\tau ,0],{\mathbb{R}}^{\ge 0}$. The operator of $\U0001d50f\left(D\right)$ involves the Riemann-Liouville fractional derivatives. In this problem, the initial conditions with fractional order and some relations among them were considered. The analysis rely on the alternative of the Leray-Schauder fixed point theorem, the Banach fixed point theorem, and the Arzela-Ascoli theorem in a cone.

##### MSC:

34A08 | Fractional differential equations |